Compared　to　the　classical　Black-Scholes　model　for　pricing　options,　the　Finite　Moment　Log　Stable　(FMLS)　model　can　more　accurately　capture　the　dynamics　of　the　stock　prices　including　large　movements　or　jumps　over　small　time　steps.　In　this　paper,　the　FMLS　model　is　written　as　a　fractional　partial　differential　equation　and　we　will　present　a　new　numerical　scheme　for　solving　this　model.　We　construct　an　implicit　numerical　scheme　with　second　order　accuracy　for　the　FMLS　and　consider　the　stability　and　convergence　of　the　scheme.　In　order　to　reduce　the　storage　space　and　computational　cost,　we　use　a　fast　bi-conjugate　gradient　stabilized　method　(FBi-CGSTAB)　to　solve　the　discrete　scheme.　A　numerical　example　is　presented　to　show　the　efficiency　of　the　numerical　method　and　to　demonstrate　the　order　of　convergence　of　the　implicit　numerical　scheme.　Finally,　as　an　application,　we　use　the　above　numerical　technique　to　price　a　European　call　option.　Furthermore,　by　comparing　the　FMLS　model　with　the　classical　B-S　model,　the　characteristics　of　the　FMLS　model　are　also　analyzed.